Calculating the standard deviation

I am performing a small experiment and I have some problems with calculating the standard deviation.

My experiment consist of an amount of measurements (N) and the result of one measurement can either be 1 or 0.

Now for instance, N=100 and the result is 1 in 50% of all measurements.

The standard deviation then:

u=0.5 (mean)

1/100*(50*(1-u)^2+50*(0-u)^2)= (50/4+50/4)/100 = 1/4.

But I want to decrease my SD. So I need to increase N. However, this doesnot have any effect. When N = 500.

1/500*(250/4+250/4)=1/4.

Obviously I am doing something wrong, can someone tell me how to calculate the SD in the correct way?